Discrete subgroups of semisimple Lie groups arise in a variety of contexts, sometimes "in nature" as monodromy groups of families of algebraic manifolds, and other times in relation to geometric structures and associated dynamical systems. I will explain a method to establish that monodromy groups of certain variations of Hodge structure give Anosov representations, thus relating algebraic and dynamical situations. Among many consequences of these interactions, I will explain some uniformization results for domains of discontinuity of the associated discrete groups, Torelli theorems for certain families of Calabi-Yau manifolds (including the mirror quintic), and also a proof of a conjecture of Eskin, Kontsevich, Möller, and Zorich on Lyapunov exponents. The discrete groups of interest live inside the real linear symplectic group, and the domains of discontinuity are inside Lagrangian Grassmanians and other associated flag manifolds. The necessary context and background will be explained.